Lecture 8

Bill Perry

Lecture 7: Review

Covered

  • Assumption tests for parametric tests
  • Statistical vs Biological significance
  • Nonparametric tests
    • Welch’s t-test: when distribution normal but variance unequal
    • Permutation test for two samples: when distribution not normal (but both groups should still have similar distributions and ~equal variance)
    • Mann-Whitney-Wilcoxon test: when distribution not normal and/or outliers are present (but both groups should still have similar distributions and ~equal variance)

Lecture 8: Overview

The objectives:

  • Decision errors
  • Data exploration and transformation
  • Exploratory graphical data analysis
  • Graphical testing of assumptions
  • Data transformation and standardization
  • Outliers

Decision errors

  • Even good studies can reach incorrect conclusions
  • “Decision errors”
  • Two types of decision errors
  • Want to know probability of making these errors

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Type I and Type II Errors

  • Type I error rate
    • α: wrongly reject H₀ when it’s true
    • α = 0.05 means a type I error rate of 5%
  • Type II error rate, β
    • wrongly fail to reject H₀ when it’s false
  • Power = 1-β: probability of correctly rejecting H₀ when H₁ is true
  • Inverse relationship between type I and type II error - but not straightforward
  • Result of chance - sample not representative of population
  • Which type of error is more dangerous?

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the dotted line is also the alpha = 0.05

Exploratory graphical data analysis

  • Graphical exploration is one of first steps in data analysis:
    • Detect data entry errors
    • Pattern exploration
    • Assess assumptions of tests
    • Detect outliers
  • Most important Q: shape of distribution?
  • Determined by density plots: “density of different values”
# Let's examine our pine needle data
# pine_data %>% 
#   group_by(wind) %>%
#   summarize(
#     n = n(),
#     mean = mean(length_mm),
#     sd = sd(length_mm),
#     min = min(length_mm),
#     max = max(length_mm)
#   )
# Histogram with density
ggplot(pine_data, aes(x = length_mm)) +
  geom_histogram(aes(y = ..density..), 
                 fill = "lightblue", 
                 color = "black",
                 bins = 10) +
  geom_density(alpha = 0.5, fill = "steelblue") +
  labs(title = "Pine Needle Length Distribution",
       x = "Length (mm)", 
       y = "Density") +
  theme_minimal()

Types of Exploratory Plots

  • Histograms: data broken into intervals, number of observations in each interval plotted on y-axis
    • Not great for small samples
# Histogram with density
ggplot(pine_data, aes(x = length_mm)) +
  geom_histogram(bins = 10) +
  labs(title = "Pine Needle Length Distribution",
       x = "Length (mm)", 
       y = "Density") +
  theme_minimal()

Types of Exploratory Plots

  • Kernel density plot: data broken into intervals, normal distribution assumed within each interval, sum of density functions plotted
# Kernel density plot
ggplot(pine_data, aes(x = length_mm)) +
  geom_density(fill = "skyblue", alpha = 0.5) +
  labs(title = "Pine Needle Length Distribution",
       x = "Length (mm)", 
       y = "Density") +
  theme_minimal()

Types of Exploratory Plots

  • Dotplots: each value represented as a dot along the measurement scale
# Dot plot of pine needle lengths
ggplot(pine_data, aes(x = 0, y = length_mm)) +
  geom_point(size = 2, alpha = 0.5,
             position = position_dodge2(width=.15)) +
  # geom_jitter(width = 0.1, height = .05, size = 2, alpha = 0.5) +
  labs(title = "Pine Needle Length Distribution",
       x = "Length (mm)", 
       y = "") +
  scale_x_continuous(limits = c(-.5, .5))+
  theme_minimal()

Types of Exploratory Plots

  • Boxplot: displays median, quartiles, range, outliers
    • Good when n > ~10

# Kernel density plot
#| message: false
#| warning: false
#| fig-height: 4
#| fig-width: 3
#| include: true
#| paged-print: false
#| 
ggplot(pine_data, aes(x = length_mm)) +
  geom_boxplot()+
  labs(title = "Pine Needle Length Distribution",
       x = "Length (mm)", 
       y = "Density") +
  theme_minimal()

Types of Exploratory Plots

  • Scatter plot: display of bivariate data
    • Shows distribution, outliers, non-linearity
  • Scatter matrix: like scatterplot, but for multiple variables -will show later


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Types of Exploratory Plots

  • QQ plots: compare quantiles of distribution against theoretical distribution (e.g. normal)
# qqplot
# QQ plot for pine needle lengths
ggplot(pine_data, aes(sample = length_mm)) +
  stat_qq() +
  stat_qq_line() +
  labs(title = "QQ Plot of Pine Needle Lengths",
       x = "Theoretical Quantiles", 
       y = "Sample Quantiles") +
  theme_minimal()

# ggplot(pine_data, aes(sample = length_mm)) +
#   stat_qq(color = "darkgreen", size = 2, alpha = 0.6) +
#   stat_qq_line(color = "blue", linewidth = 1, linetype = "dashed") +
#   labs(title = "QQ Plot of Pine Needle Lengths",
#        x = "Theoretical Quantiles", 
#        y = "Sample Quantiles") +
#   theme_minimal()

Data transformation and standardization

  • If data don’t meet distributional assumptions can try transforming:
    • Approximate a normal distribution of data and errors
    • Improve homogeneity of variance
    • Reduce effect of outliers
    • Improve linearity for regression analysis
    • Reduce interactions between variables
  • Data transformation changes the scale on which data are measured
  • Common transformations:
    • Right-skewed data: power (root) transformations, log10 transformation
    • Left-skewed data: power transformations, log10 of (constant - x)
    • Percentages/proportions (bounded): Arcsine transformation
    • Rank transformation: most extreme, leads to loss of information
# Let's apply a log transformation to our pine needle data
lt_df <- read_csv("data/lake_trout.csv")

# Let's apply a log transformation to our pine needle data
lt_df <- lt_df %>%
  mutate(log_mass = log10(mass_g +1))

# Create before and after plots to show transformation effect
lt_hist_1_plot <- ggplot(lt_df, aes(x = mass_g)) +
  geom_histogram(bins = 10, fill = "lightblue", color = "black") +
  geom_density(alpha = 0.5) +
  labs(title = "Original Data", x = "Length (mm)", y = "Count") +
  theme_minimal()

lt_qq_1_plot <- ggplot(lt_df, aes(sample = mass_g)) +
  geom_qq() + 
  geom_qq_line() +
  labs(title = "QQ Plot - Original", x = "Theoretical Quantiles", y = "Sample Quantiles") +
  theme_minimal()

lt_hist_2_log_plot <- ggplot(lt_df, aes(x = log_mass)) +
  geom_histogram(bins = 10, fill = "lightgreen", color = "black") +
  geom_density(alpha = 0.5) +
  labs(title = "Log-Transformed Data", x = "log10(Length)", y = "Count") +
  theme_minimal()

lt_qq_2_log_plot <-  ggplot(lt_df, aes(sample = log_mass)) +
  geom_qq() + 
  geom_qq_line() +
  labs(title = "QQ Plot - Log-Transformed", x = "Theoretical Quantiles", y = "Sample Quantiles") +
  theme_minimal()

# Combine plots
(lt_hist_1_plot + lt_qq_1_plot) / (lt_hist_2_log_plot + lt_qq_2_log_plot)+
  plot_annotation(
    title = "Lake Trout Mass and Log(Mass+1)"
  )

Data transformation and standardization

  • Common transformations:
    • Right-skewed data: power (root) transformations, log10 transformation
    • Left-skewed data: power transformations, log10 of (constant - x)
    • Percentages/proportions (bounded): Arcsine transformation
    • Rank transformation: most extreme, leads to loss of information

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Outliers

  • Outliers: unusual values that are outside the range of most other observations
    • Can significantly affect results of analysis
  • Outliers identified using:
    • Formal tests (Dixon’s Q, Cook’s D)
    • Graphically, using boxplots or QQ plots
  • What to do with outliers? Depends why they happened:
    • If obvious data entry error, can be removed
    • If part of the data:
      • Rerun analysis with and without outliers, report both results
      • Use tests robust to outliers or transform data
    • Unethical to remove inconvenient outliers

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Final Activity: Take home messages

Common assumptions for tests:

  1. Normality: Data comes from normally distributed populations
  2. Equal variances (for two-sample tests)
  3. Independence: Observations are independent
  4. No outliers: Extreme values can influence results

What can we do if our data violates these assumptions?

Alternatives

  • Data transformation (log, square root, etc.)
  • Non-parametric tests
  • Bootstrapping approaches

Summary and Conclusions

In this activity, we’ve:

  1. Explored decision errors (Type I and Type II) and their implications
  2. Learned various methods for exploratory data analysis
  3. Discussed data transformations to meet statistical assumptions
  4. Examined approaches for handling outliers

Key takeaways:

  • Always explore your data visually before formal analysis
  • Consider the assumptions of statistical tests and check if they are met
  • Choose appropriate transformations or alternative tests when assumptions are violated
  • Be transparent about handling outliers and report all analytical decisions

What do you see as the key points?

Things that stood out

What are the muddy points?

What does not make sense or what questions do you have…

What makes you nervous?